 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/16/a/ii/solution

Integrate the normal component on each face directly, with outward normals. Each unlisted pair of coordinates ranges independently from zero to one:

face x = 0 x = 1 y = 0 y = 1 z = 0 z = 1 v \cdot n 0 2 (y + z) - x^{2} x^{2} - 1 - y^{2} y^{2} - 1 flux 02 - 1/3 - 2/3 - 1/3 - 2/3

For example, on

x = 1

the integral is

\int_{0}^{1} \int_{0}^{1} 2 (y + z) d y d z = 2

. Adding all six contributions gives

\int_{\partial V} v \cdot d S = 0 .

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